The effects of coefficient \(({\cal G}_1 k/a)^{\prime }\)  is fully characterized by the parameter \(\nu _A\) , which leads to different damping rates of left-hand and right- hand circular polarizations of GWs, so the amplitude of left-hand circular polarization of GWs will increase (or decrease) during the propagation, while the amplitude for the right-hand modes will decrease (or increase). This effect induces an modifications to the amplitude of the GW waveform in the form of [1]}
\(h_A = h^{\rm GR}_A \exp {\left(-\frac{1}{2}\int _{\tau _e}^{\tau _0} {\cal H} \nu _A\right) d\tau } = h^{\rm GR}_A e^{\rho _A \delta h_1}, \nonumber \\\) 
